Every of L in ${\overline {K}}$ over K induces an automorphism of L.

embedding

L is the of a family of polynomials in $K[X]$ .

splitting field

Every irreducible polynomial of $K[X]$ that has a root in L splits into linear factors in L.

Let $L/K$ be an algebraic extension (i.e., L is an algebraic extension of K), such that $L\subseteq {\overline {K}}$ (i.e., L is contained in an algebraic closure of K). Then the following conditions, any of which can be regarded as a definition of normal extension, are equivalent:^[3]

If L is a normal extension of K and if E is an intermediate extension (that is, L ⊇ E ⊇ K), then L is a normal extension of E.

[4]

If E and F are normal extensions of K contained in L, then the EF and E ∩ F are also normal extensions of K.^[4]

compositum

Let L be an extension of a field K. Then:

The over K of every element in L splits in L;

minimal polynomial

There is a set $S\subseteq K[x]$ of polynomials that each splits over L, such that if $K\subseteq F\subsetneq L$ are fields, then S has a polynomial that does not split in F;

All homomorphisms $L\to {\bar {K}}$ that fix all elements of K have the same image;

The group of automorphisms, ${\text{Aut}}(L/K),$ of L that fix all elements of K, acts transitively on the set of homomorphisms $L\to {\bar {K}}$ that fix all elements of K.

Let $L/K$ be algebraic. The field L is a normal extension if and only if any of the equivalent conditions below hold.

Normal closure[edit]

If K is a field and L is an algebraic extension of K, then there is some algebraic extension M of L such that M is a normal extension of K. Furthermore, up to isomorphism there is only one such extension that is minimal, that is, the only subfield of M that contains L and that is a normal extension of K is M itself. This extension is called the normal closure of the extension L of K.

If L is a finite extension of K, then its normal closure is also a finite extension.

Galois extension

Normal basis

(2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556

Lang, Serge

(1989), Basic Algebra II (2nd ed.), W. H. Freeman, ISBN 0-7167-1933-9, MR 1009787